Problem 34 - Intermediate Algebra

Question

Perform the operations. See Example 3 . $$ -0.4(-0.6) $$

Step-by-Step Solution

Answer

The result of multiplying \(-0.4\) and \(-0.6\) is \(0.24\).

Step 1: Recognize Multiplication

The operation we are performing is multiplication between two negative decimals: \(-0.4\) and \(-0.6\). Multiplying two negative numbers results in a positive product.

Step 2: Multiply the Numbers

Ignore the signs for now and multiply the numbers as you would normally:\(0.4 \times 0.6 = 0.24\).

Step 3: Determine the Sign

Since we are multiplying two negative numbers, the result will be positive. Therefore, \(-0.4 \times -0.6 = 0.24\).

Key Concepts

Negative Numbers

Understanding negative numbers is crucial in math, especially when dealing with operations such as multiplication. A negative number is simply a number with a minus sign (-) in front of it, which indicates it is less than zero. Think of the number line: numbers to the left are negative, while numbers to the right are positive.

• Negative numbers often represent things like debt or temperature below zero.
• They follow their own rules in arithmetic operations, which are important to comprehend.

When you see two negative numbers, remember each multiplier pulls you in the opposite direction on the number line. Therefore, multiplying two negative numbers like \(-0.4\) and \(-0.6\) cancels out the negatives, resulting in a positive number.

Multiplication Rules

In math, the rules of multiplication can sometimes seem tricky, but they are essential for solving problems correctly. The primary rule to remember is:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative, and vice versa

When you multiply two numbers, ignore the signs initially. Focus first on the numbers themselves. After multiplying, apply the rule for the signs. For example, when multiplying \(0.4\) and \(0.6\), you ignore the negatives at first. The product of the two numbers is \(0.24\). Since multiplying two negative numbers results in a positive outcome, the final answer is \(+0.24\).
Understanding these rules helps solve problems quickly and efficiently with confidence.

Decimal Operations

Decimal numbers extend our number system to fractions and are essential in mathematics for precision. Performing operations with them, such as multiplication, requires care and attention to detail:

• Align the numbers based on their place value, not the decimal point.
• Ignore the decimal point initially; treat the numbers as whole numbers for multiplying.
• Once you have the product, count the total number of decimal places in the factors.
• Apply this total to the product; for example, if you multiply \(0.4\) and \(0.6\), which each have one decimal place, your product of \(0.24\) should have two decimal places.

You can see how performing operations accurately depends heavily on understanding how the process works, ensuring the outcome is as expected, precise, and correct.